Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

We extend the recent L1L1 uncertainty inequalities obtained in [13] to the metric setting. For this purpose we introduce a new class of weights, named isoperimetric weights  , for which the growth of the measure of their level sets μ({w≤r})μ({w≤r}) can be controlled by rI(r)rI(r), where I is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new localized Poincaré inequalities  , and interpolation, to prove Lp,1≤p<∞Lp,1≤p<∞, uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities of [20], and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.